Spahn
directed object' (Rev #6, changes)

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Definition (interval object)

Let CC be a closed monoidal (V-enriched for some VV) homotopical category, let ** denote the tensor unit of CC. Then the cospan category (same objects and cospans as morphisms) is VV-enriched, too.

An interval object is defined to be a cospan *aIb**\stackrel{a}{\rightarrow}I\stackrel{b}{\leftarrow}*.

The pushout I 2I^{\coprod_2} of this diagram satisfies *[I,I 2] *{}_*[ I,I^{\coprod_2}]_*\simeq is contractible (see co-span for this notation).

Definition (directed object)

Let VV be a monoidal category, let CC be VV-enriched, closed monoidal homotopical category, let ** denote the tensor unit of CC which we assume to be the terminal object, let *0I1*\stackrel{0}{\to} I\stackrel{1}{\leftarrow} denote the interval object of CC, let XX be a pointed object of CC. Let DD be the functor bifunctorD:C×CV D:C\to D:C\times C\to V, (X,I)[I,X] X\mapsto (X,I)\mapsto [I,X].

A direction in CC is defined to be a subfunctor dd of D(I,) D D(I,-) preserving the terminal object and satisfying the properties below. In this case dXd X is called a direction for XX. A global element of dXdX is called a dd-directed path in XX and their collection we denote by ddp(X)ddp(X) . the The properties are:

(1) The DD image of every map I*XI\to *\to X factoring over the point is in ddp(X)ddp(X).

(2) ddp f ( , X g )dX ddp(X) f,g\in dX is are closed called under to the be tensor a product. Forα,βddp(X)\alpha,\beta\in ddp(X)composable pair , their if pushoutαf¯β0=g¯1 \alpha\otimes \overline \beta f\circ 0=\overline g\circ 1 where the overlined letters denote the adjoints under the hom adjunction. The composition of a composable pair is called defined their bycompositionfg:=f¯g¯̲f\circ g:=\underline{\overline{f}\bullet\overline{g}} . Then the condition is thatddp(X)ddp(X) is closed under composition.

A (3) Ifdirected objectτhom(I,I)\tau\in hom (I,I) is and defined to be a pair dfX:ddp=(X,dX) {}_d f\in X:=(X,dX) ddp(X) consisting then of an objectXf¯τ̲dX X \underline{\overline{f}\circ \tau}\in dX of where the underlined term denotes the adjoint (in the other direction) under the hom adjunction.CC and a direction dXdX for XX.

A (4)morphism of directed objectsddp(X)ddp(X) is a functorf:(X,dX)(Y,dy)f:(X,dX)\to (Y,dy) is defined to be a pair (f,df)(f,df) making

X f Y d d dX df dY\array{ X& \stackrel{f}{\to}& Y \\ \downarrow^d&&\downarrow^d \\ dX& \stackrel{df}{\to}& d Y }

A directed object is defined to be a pair dX:=(X,dX){}_d X:=(X,dX) consisting of an object XX of CC and a direction dXdX for XX.

A morphism of directed objects f:(X,dX)(Y,dY)f:(X,dX)\to (Y,dY) is defined to be a pair (f,df)(f,df) making

Remark

Let CC be VV-enriched, closed monoidal homotopical category, let II denote the tensor unit of CC.

Definition

A directed-path-space objects is defined.

Revision on November 6, 2012 at 00:41:31 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.